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REVIEW 3 major objections 4 minor 36 references

Emptiness Instanton in Quantum Polytropic Gas

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Pith's one-line read This paper proves that the potential $V_n$ defined by a Pochhammer contour integral solves the emptiness instanton problem for every polytropic index $\gamma > 1$, confirming the conjectured emptiness formation probability exponent $f(n)$…

desk verdict Proves the conjectured emptiness exponent for arbitrary polytropic index via a neat analytic continuation, but the admitted branch mismatch leaves the full spacetime profile and the x=-1 endpoint unproven. read the letter →

arxiv 2412.11686 v3 pith:PUVHHUKZ submitted 2024-12-16 cond-mat.stat-mech math-phmath.MPnlin.PSquant-ph

Alexander G. Abanov , Dimitri M. Gangardt This is my paper · ORCID
classification cond-mat.stat-mechmath-phmath.MPnlin.PSquant-ph
keywords emptinessformationprobabilitypolytropicgasinstantonhydrodynamicsinimaginarytimePochhammercontourintegralDotsenko-FateevintegralsEuler-Poissonequationlargedeviations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper addresses the emptiness formation problem: how likely it is that a macroscopically empty interval spontaneously appears in the ground state of a one-dimensional quantum gas with a polytropic equation of state $P \sim \rho^{\gamma}$. For a long interval, the probability is exponentially small, $P_{\mathrm{EFP}} \sim e^{-(R^2 \rho_0 m c_0 / \hbar) f(n)}$, and is dominated by a classical instanton solution of the hydrodynamic equations in imaginary time. Previous work obtained the instanton only for integer $n$ and conjectured the closed form $f(n) = \frac{2}{n+1}\left[\frac{\Gamma(n+3/2)}{\Gamma(n+1)}\right]^2$ for all $n$. This paper constructs the potential $V_n(\lambda, \bar{\lambda})$ through a Pochhammer contour integral, verifies the Euler-Poisson equation, the endpoint boundary conditions, and the quadrupole asymptotic amplitude, and thereby proves the instanton and the exponent $f(n)$ for all $n > -1/2$, that is, for all $\gamma > 1$. The result makes the emptiness probability analytically known across a continuous range of polytropic indices, including the weakly interacting Bose gas at $\gamma = 2$ ($n = 1/2$), which was previously accessible only numerically.

What carries the argument

The load-bearing object is the potential $V_n(\lambda,\bar{\lambda})$ appearing in the hodograph equation $x - w\tau = \partial_\lambda V_n$ and its complex conjugate. For integer $n$ this potential was a finite sum; here it is written as a contour integral, Eq. (34), whose integrand contains $z(z^2+1)^{n-1/2} / [(z-\lambda)^n(z-\bar{\lambda})^n]$ integrated around a Pochhammer contour that winds around the branch points $i$ and $-i$ in opposite senses. That contour is what makes the analytic continuation to non-integer $n$ single-valued. The same integrand satisfies the Euler-Poisson equation (21) for every $z$, so $V_n$ inherits it by linearity; then the boundary condition at the interval endpoints and the quadrupole asymptotics are extracted from the equivalent one-dimensional integrals (36) and (43). From this machinery the paper derives the empty-region boundary via the function $g(v)$ in Eq. (50) and the explicit singularity exponents of the density at the axes.

What would settle it

Compute the density profile along a curve in the $(x,\tau)$ plane that crosses the boundary $\rho = 1$ away from the axes using both Eq. (36) and Eq. (43); if the two branches disagree there, the off-axis part of the instanton and the claimed exponent at $x = \pm 1$ are unsupported.

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Extended reading notes

Core claim

The central claim is that Eq. (34), the potential $V_n(\lambda, \bar{\lambda})$ defined by a Pochhammer contour integral together with the prefactor $2/(1+e^{2\pi i n})^2$, is the correct analytic continuation of the emptiness instanton of Ref. [3] to non-integer $n$, and is valid for every real $n > -1/2$ (equivalently $\gamma > 1$). The argument proceeds in three steps: the integrand obeys the Euler-Poisson equation (21) at every point $z$, so $V_n$ satisfies it by linearity; the collapsed-contour representation (36) gives $\partial_\lambda V_n \to \pm 1$ as $|\lambda| \to \infty$, matching the required singularities at the interval endpoints; and the evaluation (38) on the symmetry line $\lambda = -\bar{\lambda} = i\mu$ reproduces the inverse-square-root quadrupole asymptotics (24) with amplitude $\alpha = \frac{1}{2}(2n+1)\left[\frac{\Gamma(n+3/2)}{\Gamma(3/2)\Gamma(n+1)}\right]^2$, which feeds into $f(n)$ through Eq. (25). Thus the emptiness formation probability exponent (3), the critical time (4), and the spatiotemporal instanton profile (41)-(42) hold for arbitrary polytropic index $\gamma > 1$, not just the integer values treated before.

Load-bearing premise

The two integral representations used for the plotted profile must give the same branch of the multi-valued potential in the overlap region where the density crosses $\rho = 1$; the paper asserts but does not prove their equivalence there.

Editorial extensions

If this is right

  • The emptiness formation probability exponent is now proven in closed form for all $\gamma > 1$: $f(n) = \frac{2}{n+1}\left[\frac{\Gamma(n+3/2)}{\Gamma(n+1)}\right]^2$, so the exponential suppression of emptiness is known for every polytropic index.
  • The analytic continuation includes the weakly interacting Bose gas case $\gamma = 2$, $n = 1/2$, for which the instanton profile was previously obtained only by numerical solution of the hydrodynamic equations.
  • The spacetime shape of the empty region is astroid-like for all $n$; near $x=0$, $\tau = \pm\tau_c$ the boundary always scales as $|x| \sim (|\tau|-\tau_c)^{3/2}$, while near $x = \pm 1$, $\tau = 0$ the scaling exponent is $n$-dependent, $|\tau| \sim (1-|x|)^{(2n+3)/(2n+2)}$.
  • Explicit density profiles are available on the symmetry axes: $\rho(0,\tau) = [1 - (\tau_c/\tau)^2]^{n+1/2}$ and $\rho(x,0) \sim (A_n/(|x|-1))^{(2n+1)/(2n+2)}$ near the endpoint.
  • The closed-form hydrodynamic profile provides concrete predictions that can be checked by direct numerical simulation of the imaginary-time hydrodynamic equations for any $\gamma > 1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the branch-patching between Eqs. (36) and (43) is benign on the overlap, the same Pochhammer-contour construction should produce exact instanton solutions for other large-deviation observables in polytropic gases, such as full counting statistics, by changing only the boundary conditions.
  • The Dotsenko-Fateev form of the potential hints that emptiness formation in an interacting gas may be governed by a conformal field theory with a $\gamma$-dependent central charge; if so, exact microscopic universality would extend beyond free fermions.
  • The coexistence of a universal $3/2$ exponent at $\tau = \pm\tau_c$ with $n$-dependent exponents at $x = \pm 1$ suggests that fluctuations around the emptiness boundary may exhibit an $n$-dependent dynamical exponent, a question the authors explicitly leave open and that could be tested by instanton fluctuation calculations.
  • Formally continuing to $n = -1$ (Chaplygin gas) makes $\tau_c = f(n) = 0$; the comment in the paper that this continuation solves a different problem can be tested by checking whether the analytic potential (34) at $n = -1$ yields a real, positive hydrodynamic density profile at all.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper constructs a Pochhammer contour integral, Eq. (34), for the potential V_n governing the imaginary-time hydrodynamic description of a one-dimensional polytropic gas with equation of state P ~ rho^gamma, gamma = 1 + 2/(2n+1), and claims that this potential solves the emptiness instanton problem for all n > -1/2. The construction verifies the Euler-Poisson equation by linearity, checks the endpoint and quadrupole boundary conditions using Eq. (36) and hypergeometric asymptotics, and rederives the emptiness formation amplitude alpha in Eq. (39), and therefore the EFP exponent f(n) in Eq. (3), for non-integer n. The paper then uses Eqs. (41)-(42) to compute the spatiotemporal profile in Fig. 1, including the astroid-shaped empty region, the critical time Eq. (4), and n-dependent singular exponents near x = ±1 and tau = ±tau_c.

Significance. If the construction is fully established, this is a substantial result: it gives the first analytic emptiness instanton for non-integer polytropic indices, proves the conjectured EFP exponent of Ref. [3] for all gamma > 1, and connects the hydrodynamic limit-shape problem to Dotsenko-Fateev integrals. The paper has genuine strengths: there are no fitted parameters; the Euler-Poisson equation is satisfied by linearity; the boundary conditions and the quadrupole amplitude are checked through explicit hypergeometric manipulations; and the amplitude reproduces the integer-n result exactly. The result would be of interest to the statistical mechanics and cold-atom communities. However, the global spatiotemporal profile claim is currently weakened by an admitted branch ambiguity in the patching of the two integral representations, so the paper needs revision before the full claim can be accepted.

major comments (3)
  1. [Section 5 and Appendix B, Eqs. (36), (43), and Fig. 1] The full spatiotemporal profile is obtained by patching two integral representations whose equivalence is explicitly not guaranteed. The derivation of Eq. (36) and the identity leading to Eq. (43) assume Re lambda = Re bar lambda = v > 0, as stated in Appendix B. The physical profile includes the left half of the astroid, where v < 0, and the density crosses rho = 1 where the text switches between Eq. (36) and Eq. (43). Appendix B closes by stating that outside v > 0 the equivalence between the two representations 'can be violated.' Consequently the boundary condition (22) at x = -1, the asymptotic density (47), and the boundary scaling (56) are not established for the continued solution. The EFP exponent obtained from Eq. (38) on the axis is less exposed, but the global instanton claim requires either a branch-continuation proof or a direct numerical check that both representations agree on the overlap and on the v < 0 part of the physical domain.
  2. [Sections 1 and 5, gamma = 2 case] The manuscript highlights gamma = 2, n = 1/2, as a case studied numerically in Ref. [2], but it contains no quantitative comparison with that numerical solution. Since the new step is analytic continuation in n, a comparison of at least rho(x,0), rho(0,tau), and the boundary shape with the numerical data of Ref. [2] is the natural falsifiable check of the branch choice. Without it, the claim that the construction correctly covers the weakly interacting Bose gas rests on internal consistency alone.
  3. [Section 4, Eq. (34)] The prefactor 2/(1 + e^{2 pi i n})^2 in Eq. (34) is singular at n = 1/2, which is the value corresponding to gamma = 2 and is displayed in Fig. 1. The text explains the factor only for integer n, where it equals 1/2. If the Pochhammer integral vanishes at the same values so that the limit is finite, that cancellation should be shown explicitly; otherwise Eq. (34) is not a valid representation for all n > -1/2 as claimed.
minor comments (4)
  1. [Section 5, Eq. (46)] The stated expansion near x = 0, |tau| -> tau_c^+ is not the expansion of Eq. (45); the correct leading behavior is (2(|tau| - tau_c)/tau_c)^(n + 1/2), not ((2|tau| - tau_c)/tau_c)^(n + 1/2).
  2. [Section 5, Eq. (43)] The numerical evaluation of Eq. (43) uses a principal-value prescription at q = 0; the implementation should be described briefly so that the profiles in Fig. 1 are reproducible and it is clear how the branch switch between Eq. (36) and Eq. (43) is implemented in practice.
  3. [Figure 1 caption] The caption should indicate which region of the profile is computed with Eq. (36) and which with Eq. (43), since the switching between representations is part of the construction and is relevant to interpreting the plotted density.
  4. [Appendix A and Section 5] There are minor typographical errors, such as 'indpendent' in Appendix A and 'densties' in Section 5, which should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the contour-integral potential is independently verified, and self-citations to Ref. [3] provide the integer-n starting point and the action-amplitude relation, not the generalized EFP result.

full rationale

The paper's derivation is not circular in the relevant sense. The potential V_n is defined by the Pochhammer contour integral, Eq. (34), and the authors verify, rather than assume, that it satisfies the required conditions: the Euler-Poisson equation (21) is checked by linearity of the integrand, the boundary condition (22) is obtained from the collapsed integral representation (36) for n > -1/2, and the quadrupole amplitude alpha in Eq. (39) is computed from the hypergeometric expansion of Eq. (38), an independent mathematical step. The EFP exponent f(n) then follows from the cited relation (25) of Ref. [3], but alpha is not fitted and f(n) is not used as an input; the cited relation is a parameter-free general relation between the amplitude and the action, so it is independent support under the stated rules. The self-citations to Ref. [3] are legitimate prior work: they supply the integer-n construction that is being analytically continued and the conjectured target formula, but the present paper rederives the amplitude for arbitrary n rather than importing the final exponent. The branch-consistency concern raised in Appendix B, where the equivalence of the representations in Eqs. (36) and (43) 'can be violated' outside Re lambda = Re bar-lambda = v > 0, is a correctness or completeness issue for the full spatiotemporal profile, not a circularity: it does not show that any quantity is being defined in terms of its own prediction. No fitted parameters, no prediction that reduces to a fit, and no load-bearing uniqueness theorem imported from the authors' previous work were found. The overall circularity score is therefore low, reflecting only the presence of normal self-citation in a derivation that is otherwise self-contained.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central result depends on the validity of the hydrodynamic effective action (domain assumption), the semiclassical instanton approximation, and the specific Pochhammer-contour analytic continuation of V_n, which is verified for boundary conditions but not proven globally. No free parameters or new physical entities are introduced.

assumptions (6)
  • domain assumption The 1D quantum gas is described by an effective hydrodynamic action with polytropic equation of state P ~ rho^gamma (Eqs. 8-10).
    The paper takes this zero-temperature hydrodynamic description as the starting point; the microscopic derivation of this action is not provided.
  • domain assumption Emptiness formation probability is dominated by a single classical instanton solution in the large-R limit (semiclassical saddle point, Eq. 13).
    Standard instanton calculus; the paper assumes this rather than deriving fluctuations around the saddle point.
  • domain assumption The Wick-rotated imaginary-time hydrodynamics and the boundary conditions (22) and (24) uniquely characterize the emptiness instanton.
    The solution is constructed to satisfy these conditions; uniqueness is not proved.
  • ad hoc to paper The Pochhammer contour integral representation (34) is the correct analytic continuation of the integer-n solution and satisfies the required parity and regularity conditions (prefactor factor ensures d_lambda V(0,0)=0).
    This is the central construction of the paper; it is verified for the boundary conditions (22) and (24) but not rigorously justified as the unique continuation.
  • standard math Standard complex analysis results: residue theorem, Pochhammer contour properties, hypergeometric identities such as DLMF 15.4.23, and gamma-function identities.
    Used in deriving Eqs. (36)-(39) and the boundary asymptotics (44)-(53).
  • domain assumption For the integral manipulations, the domain Re lambda = Re \bar lambda = v > 0 fixes the branch; results are assumed to remain valid where the paper uses them.
    Appendix B notes equivalence of the representations can be violated outside this domain, so the profiles rely on a domain assumption.

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Pith. "Pith review of Emptiness Instanton in Quantum Polytropic Gas." pith.science (2026). https://pith.science/paper/PUVHHUKZ

@misc{pith2026241211686,
  author       = {Pith},
  title        = {Pith review of: Emptiness Instanton in Quantum Polytropic Gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUVHHUKZ}},
  note         = {Machine review of arXiv:2412.11686}
}
abstract

The emptiness formation problem is addressed for a one-dimensional quantum polytropic gas characterized by an arbitrary polytropic index $\gamma$, which defines the equation of state $P \sim \rho^\gamma$, where $P$ is the pressure and $\rho$ is the density. The problem involves determining the probability of the spontaneous formation of an empty interval in the ground state of the gas. In the limit of a macroscopically large interval, this probability is dominated by an instanton configuration. By solving the hydrodynamic equations in imaginary time, we derive the analytic form of the emptiness instanton. This solution is expressed as an integral representation analogous to those used for correlation functions in Conformal Field Theory. Prominent features of the spatiotemporal profile of the instanton are obtained directly from this representation.

Figures

Figures reproduced from arXiv: 2412.11686 by the authors.

Figure 1
Figure 1. Emptiness instanton spatiotemporal density profiles for various polytropic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Left: Temporal extension of the emptiness instanton, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Left: contours C (λ,λ¯) and C (i,−i) in Eqs (32) and(33). Right: Pochham￾mer contour Πoχ in Eqs (34) and (35). the series (27) can be represented in a rather compact way as a multiple derivative, Vn (λ,λ¯) = 1 n! ∂ n−1 λ λ(λ 2 + 1) n− 1 2 (λ − λ¯) n + c.c. . (31) This representation is still limited to positive integer values of n. The crucial observation leading to the results in this paper is that for these values… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Left: Density profile at τ = 0. Right: Density profile at x = 0 given by Eq. (45) 6 Conclusions and Open Questions Our main achievement is determining the Emptiness Formation Probability (EFP) resulting from a large quantum fluctuation in a one-dimensional gas with a p…

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Works this paper leans on

36 extracted references · 26 canonical work pages

  1. [3]

    H.-C. Yeh, D. M. Gangardt and A. Kamenev, Emptiness formation in polytropic quantum liquids, J. Phys. A 55(6), 064002 (2022), doi:10.1088 /1751-8121/ac47b1. 13 SciPost Physics Submission

    work page 2022

  2. [2]

    Yeh and A

    H.-C. Yeh and A. Kamenev, Emptiness formation probability in one-dimensional Bose liq- uids, Phys. Rev. A 101, 023623 (2020), doi:10.1103 /PhysRevA.101.023623

    work page 2020

  3. [1]

    Arzamasovs and D

    M. Arzamasovs and D. M. Gangardt, Full counting statistics and large de- viations in a thermal 1d Bose gas , Phys. Rev. Lett. 122, 120401 (2019), doi:10.1103/PhysRevLett.122.120401

    doi:10.1103/physrevlett.122.120401 2019

  4. [4]

    Doyon, G

    B. Doyon, G. Perfetto, T . Sasamoto and T . Yoshimura,Emergence of hydrodynamic spa- tial long-range correlations in nonequilibrium many-body systems , Phys. Rev. Lett. 131, 027101 (2023), doi:10.1103 /PhysRevLett.131.027101

    work page 2023

  5. [5]

    McCulloch, J

    E. McCulloch, J. De Nardis, S. Gopalakrishnan and R. Vasseur, Full counting statistics of charge in chaotic many-body quantum systems , Phys. Rev. Lett. 131, 210402 (2023), doi:10.1103/PhysRevLett.131.210402

    work page doi:10.1103/physrevlett.131.210402 2023

  6. [6]

    J. S. Pallister, S. H. Pickering, D. M. Gangardt and A. G. Abanov, Phase transitions in full counting statistics of free fermions and directed polymers , arXiv:2405.12651 (2024), doi:10.48550/arXiv.2405.12651

    work page Pith review arXiv doi:10.48550/arxiv.2405.12651 2024

  7. [7]

    Gopalakrishnan, A

    S. Gopalakrishnan, A. Morningstar, R. Vasseur and V . Khemani, Distinct universality classes of diffusive transport from full counting statistics, Phys. Rev. B109, 024417 (2024), doi:https://doi.org/10.1103/PhysRevB.109.024417

    doi:10.1103/physrevb.109.024417 2024

  8. [8]

    W . S. Bakr, J. I. Gillen, A. Peng, S. Fölling and M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice , Nature 462, 74 (2009), doi:10.1038/nature08482

    doi:10.1038/nature08482 2009

Show all 36 references
  1. [9]

    J. F . Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch and S. Kuhr, Single- atom-resolved fluorescence imaging of an atomic Mott insulator , Nature 467, 68 (2010), doi:10.1038/nature09378

    2010 doi

  2. [10]

    Haller, J

    E. Haller, J. Hudson, A. Kelly , D. A. Cotta, B. Peaudecerf, G. D. Bruce and S. Kuhr,Single- atom imaging of fermions in a quantum-gas microscope, Nature Physics 11, 738 (2015), doi:10.1038/nphys3403

    2015 doi

  3. [11]

    M. F . Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif and M. Greiner, Site-resolved measurement of the spin-correlation function in the Fermi-Hubbard model , Science 353, 1253 (2016), doi:10.1126 /science.aag1430

    2016

  4. [12]

    Gross and I

    C. Gross and I. Bloch,Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017), doi:10.1126 /science.aal3837

    2017

  5. [13]

    Mitra, P

    D. Mitra, P . T . Brown, E. Guardado-Sanchez, S. S. Kondov, T . Devakul, D. A. Huse, P . Schauß and W . S. Bakr,Quantum gas microscopy of an attractive Fermi–Hubbard system, Nature Physics 14, 173 (2018), doi:10.1038 /nphys4297

    2018

  6. [14]

    D. Wei, A. Rubio-Abadal, B. Ye, F . Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch and J. Zeiher,Quantum gas microscopy of Kardar- Parisi-Zhang superdiffusion, Science 376, 716 (2022), doi:10.1126 /science.abk2397

    2022

  7. [15]

    Stéphan, Extreme boundary conditions and random tilings, SciPost Physics Lecture Notes (2021), doi:10.21468 /scipostphyslectnotes.26

    J.-M. Stéphan, Extreme boundary conditions and random tilings, SciPost Physics Lecture Notes (2021), doi:10.21468 /scipostphyslectnotes.26

    2021

  8. [16]

    Kenyon, Lectures on dimers, arXiv:0910.3129 (2009), doi:10.48550/arXiv.0910.3129

    R. Kenyon, Lectures on dimers, arXiv:0910.3129 (2009), doi:10.48550/arXiv.0910.3129

    2009 doi arXiv

  9. [17]

    Colomo and A

    F . Colomo and A. G. Pronko,The arctic curve of the domain-wall six-vertex model, J. Stat. Phys. 138, 662 (2010), doi:10.1007 /s10955-009-9902-2

    2010

  10. [18]

    Colomo, A

    F . Colomo, A. G. Pronko and P . Zinn-Justin,The arctic curve of the domain wall six-vertex model in its antiferroelectric regime , J. Stat. Mech. Theory Exp. 2010, L03002 (2010), doi:10.1088/1742-5468/2010/03/L03002. 14 SciPost Physics Submission

    2010 doi

  11. [19]

    Colomo and A

    F . Colomo and A. Sportiello, Arctic Curves of the Six-Vertex Model on Generic Domains: The Tangent Method, J. Stat. Phys. 164, 1488 (2016), doi:10.1007/s10955-016-1590-0

    2016 doi

  12. [20]

    Borodin, I

    A. Borodin, I. Corwin and V . Gorin,Stochastic six-vertex model, Duke Math. J. 165, 563 (2016), doi:10.1215 /00127094-3166843

    2016

  13. [21]

    Reshetikhin and A

    N. Reshetikhin and A. Sridhar, Limit shapes of the stochastic six vertex model , Commun. Math. Phys. 363, 741 (2018), doi:10.1007 /s00220-018-3253-2

    2018

  14. [22]

    Burenev, F

    I. Burenev, F . Colomo, A. Maroncelli and A. Pronko,Arctic curves of the four-vertex model, J. Phys. A 56, 465202 (2023), doi:10.1088 /1751-8121/ad02ce

    2023

  15. [23]

    de Gier, R

    J. de Gier, R. Kenyon and S. S. Watson,Limit shapes for the asymmetric five vertex model, Commun Math. Phys. 385, 793 (2021), doi:10.1007 /s00220-021-04126-7

    2021

  16. [24]

    A. G. Abanov, Hydrodynamics of correlated systems , In É. Brézin, V . Kazakov, D. Serban, P . Wiegmann and A. Zabrodin, eds., Applications of Random Matrices in Physics, pp. 139–161. Springer Netherlands, Dordrecht, ISBN 978-1-4020-4531-8, doi:10.48550/arXiv.cond-mat/0504307 (...

    2006 doi arXiv

  17. [25]

    A. G. Abanov and V . E. Korepin,On the probability of ferromagnetic strings in antiferromag- netic spin chains, Nucl. Phys. B 647, 565 (2002), doi:10.1016/S0550-3213(02)00899-4

    2002 doi

  18. [26]

    Joseph, J

    J. Joseph, J. E. Thomas, M. Kulkarni and A. G. Abanov, Observation of shock waves in a strongly interacting fermi gas , Phys. Rev. Lett. 106, 150401 (2011), doi:10.1103/PhysRevLett.106.150401

    2011 doi

  19. [27]

    R. Jackiw, A particle field theorist’s lectures on supersymmetric, non-abelian fluid mechanics and d-branes , arXiv preprint physics /0010042 (2000), doi:10.48550/arXiv.physics/0010042

    2000 doi arXiv

  20. [28]

    E. T . Whittaker and G. N. Watson,A Course of Modern Analysis, Cambridge Mathematical Library . Cambridge University Press, 4 edn. (1996)

    1996

  21. [29]

    NIST Digital Library of Mathematical Functions, https: //dlmf.nist.gov/, Release 1.2.1 of 2024-06-15, F . W . J. Olver, A. B. Olde Daalhuis, D. W . Lozier, B. I. Schneider, R. F . Boisvert, C. W . Clark, B. R. Miller, B. V . Saunders, H. S. Cohl, and M. A. McClain, eds

    2024

  22. [30]

    Dotsenko and V

    V . Dotsenko and V . Fateev,Conformal algebra and multipoint correlation functions in 2d statistical models, Nucl. Phys. B 240, 312 (1984), doi:10.1016/0550-3213(84)90269-4

    1984 doi

  23. [31]

    Dotsenko and V

    V . Dotsenko and V . Fateev,Four-point correlation functions and the operator algebra in 2d conformal invariant theories with central charge c ≤ 1, Nucl. Phys. B 251, 691 (1985), doi:10.1016/S0550-3213(85)80004-3

    1985 doi

  24. [32]

    Di Francesco, P

    P . Di Francesco, P . Mathieu and D. Sénéchal,Conformal field theory, Graduate Texts in Contemporary Physics. Springer, Germany , ISBN 0-387-94785-X, doi:10.1007 /978-1- 4612-2256-9 (1997)

    1997

  25. [33]

    J. S. Pallister, D. M. Gangardt and A. G. Abanov, Limit shape phase transitions: a merger of arctic circles, J. Phys. A 55, 304001 (2022), doi:10.1088 /1751-8121/ac79ad

    2022

  26. [34]

    Allegra, J

    N. Allegra, J. Dubail, J.-M. Stéphan and J. Viti, Inhomogeneous field theory inside the arctic circle , J. Stat. Mech. Theory Exp. 2016, 053108 (2016), doi:10.1088 /1742- 5468/2016/05/053108. 15 SciPost Physics Submission

    2016

  27. [35]

    Dubail, J.-M

    J. Dubail, J.-M. Stéphan, J. Viti and P . Calabrese, Conformal Field Theory for Inhomo- geneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases, SciPost Phys. 2, 002 (2017), doi:10.21468 /SciPostPhys.2.1.002

    2017

  28. [36]

    A. M. Kamchatnov, Nonlinear periodic waves and their modulations: an introductory course, World Scientific, Singapore (2000). A Hydrodynamics of Polytropic Gas Hydrodynamic configurations are specified by density ,ρ(x, t), and velocity v(x, t) fields obey- ing continuity equat...

    2000

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.